Notes

Symbolic systems of the general type I discuss here seem to have first arisen in 1920 in the work of Moses Schönfinkel on what became known as combinators. As discussed on page 1121 Schönfinkel introduced certain specific rules that he suggested could be used to build up functions defined in logic. Beginning in the 1930s there were a variety of theoretical studies of how logic and mathematics could be set up with combinators, notably by Haskell Curry. For the most part, however, only Schönfinkel's specific rules were ever used, and only rather specific forms of behavior were investigated. In the 1970s and 1980s there was interest in using combinators as a basis for compilation of functional programming languages, but only fairly specific situations of immediate practical relevance were considered. (Combinators have also been used as logic recreations, notably by Raymond Smullyan.)

Constructs like combinators appear to have almost never been studied in mainstream pure mathematics. Most likely the reason is that building up functions on the basis of the structure of symbolic expressions has never seemed to have much obvious correspondence to the traditional mathematical view of functions as mappings. And in fact even in mathematical logic, combinators have usually not been considered mainstream. Most likely the reason is that ever since the work of Bertrand Russell in the early 1900s it has generally been assumed that it is desirable to distinguish a hierarchy of different types of functions and objects—analogous to the different types of data supported in most programming languages. But combinators are set up not to have any restrictions associated with types. And it turns out that among programming languages Mathematica is almost unique in also having this same feature. And from experience with Mathematica it is now clear that having a symbolic system which—like combinators—has no built-in notion of types allows great generality and flexibility. (One can always set up the analog of types by having rules only for expressions whose heads have particular structures.)